Problem: All of the 3rd grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$6.00$ each for teachers and $$3.50$ each for students, and the group paid $$29.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$24.00$ each for teachers and $$12.50$ each for students, and the group paid $$110.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+3.5y = 29.5}$ ${24x+12.5y = 110.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-24x-14y = -118}$ ${24x+12.5y = 110.5}$ Add the top and bottom equations together. $ -1.5y = -7.5 $ $ y = \dfrac{-7.5}{-1.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {6x+3.5y = 29.5}$ to find $x$ ${6x + 3.5}{(5)}{= 29.5}$ $6x+17.5 = 29.5$ $6x = 12$ $x = \dfrac{12}{6}$ ${x = 2}$ You can also plug ${y = 5}$ into $ {24x+12.5y = 110.5}$ and get the same answer for $x$ ${24x + 12.5}{(5)}{= 110.5}$ ${x = 2}$ There were $2$ teachers and $5$ students on the field trips.